A minicourse on stochastic partial differential equations / Robert Dalang ... [and others] ; editors, Davar Khoshnevisan, Firas Rassoul-Agha.

Format:

Book

Contributor:

Dalang, Robert C., 1961-

Khoshnevisan, Davar.

Rassoul-Agha, Firas.

Series:

Lecture notes in mathematics (Springer-Verlag) ; 1962.

Lecture notes in mathematics, 0075-8434 ; 1962

Language:

English

Subjects (All):

Stochastic partial differential equations.

Stochastic differential equations.

Physical Description:

xi, 216 pages ; 24 cm.

Place of Publication:

Berlin : Springer, [2009]

Summary:

In May 2006, The University of Utah hosted an NSF-funded minicourse on stochastic partial differential equations. The goal of this minicourse was to introduce graduate students and recent Ph.D.s to various modern topics in stochastic PDEs, and to bring together several experts whose research is centered on the interface between Gaussian analysis, stochastic analysis, and stochastic partial differential equations. This monograph contains an up-to-date compilation of many of those lectures. Particular emphasis is paid to showcasing central ideas and displaying some of the many deep connections between the mentioned disciplines, all the time maintaining a realistic pace for the student of the subject.

Contents:

A Primer on Stochastic Partial Differential Equations / Davar Khoshnevisan 1

1 What is an SPDE? 1

2 Gaussian Random Vectors 2

3 Gaussian Processes 2

4 Regularity of Random Processes 8

5 Martingale Measures 14

6 A Nonlinear Heat Equation 23

7 From Chaos to Order 32

The Stochastic Wave Equation / Robert C. Dalang 39

2 The Stochastic Wave Equation 41

3 Spatially Homogeneous Gaussian Noise 47

4 The Wave Equation in Spatial Dimension 2 49

5 A Function-Valued Stochastic Integral 56

6 The Wave Equation in Spatial Dimension d >= 1 58

7 Spatial Regularity of the Stochastic Integral (d = 3) 61

8 Holder-Continuity in the 3-d Wave Equation 70

Application of Malliavin Calculus to Stochastic Partial Differential Equations / David Nualart 73

2 Malliavin Calculus 73

3 Application of Malliavin Calculus to Regularity of Probability Laws 83

4 Stochastic Heat Equation 92

5 Spatially Homogeneous SPDEs 99

Some Tools and Results for Parabolic Stochastic Partial Differential Equations / Carl Mueller 111

2 Basic Framework 113

3 Duality 115

4 Large Deviations for SPDEs 125

5 A Comparison Theorem 129

6 Applications 131

Sample Path Properties of Anisotropic Gaussian Random Fields / Yimin Xiao 145

2 Examples and General Assumptions 148

3 Properties of Strong Local Nondeterminism 160

4 Modulus of Continuity 164

5 Small Ball Probabilities 168

6 Hausdorff and Packing Dimensions of the Range and Graph 170

7 Hausdorff Dimension of the Level Sets and Hitting Probabilities 183

8 Local Times and Their Joint Continuity 194.

Notes:

"From May 8 to May 19th of 2006, the Department of Mathematics at the University of Utah hosted a minicourse on some modern topics in stochastic partial differential equations ... The present book is comprised of most of those lectures"--Pref.

Includes bibliographical references and index.

ISBN:

9783540859932

3540859934

OCLC:

264694035